CPA SCHOOL OF WASHINGTON
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DIVERSIFICATION
Should you avoid putting all your eggs in one basket or is it better, in the words of one famous captain of industry, to put all your eggs in one basket and watch the basket carefully?
Let us start with two premises:
- You like your returns to be as high as possible; and
- you dislike risk, you are "risk averse."
When you invest in a security, piece of land, new venture, you like your return on your investment to be as high as possible, preferably double digit or even triple digit. At the same time, however, you would like to be sure of that return and not have to worry that it will be much less.
By "risk averse" it will be meant that you do not like your return on investment to have large "ups" and large "downs." Needless to say, you don't really mind "ups," it is the "downs" you don't like. Nevertheless, it is convenient to measure risk in terms of both ups and downs. Thus, your desire for a high return or a high rate of return is usually expressed in terms of a desire for a high "expected value" or "most likely value" while your risk is usually expressed in terms of some measure of the dispersion or possible "scatter" of the actual returns from the expected or most likely value.
Thus a high tech stock might seem to promise a high rate of return but might also have a high degree of risk because the actual return might be far below or far above that expected high rate of return. By contrast, a U.S. Treasury security might have the undesirable feature that the return you expect from it might be quite low but it nevertheless has the desirable feature that there is little doubt about what that return will be.
In the next several pages, the argument will be made that it is generally a good idea not to put all your eggs in one basket. If the benefits of diversification seem self evident, you will find it nevertheless interesting the read this development. You will find, for example, that investing all your resources in one very low risk security (say a long term Treasury bond) will not only have less return but also potentially greater risk than investing those same resources in a broad portfolio of Treasury securities, corporate debt, and possibly some common stock and real estate.
Perhaps you may say that surely investing all the portfolio in a long-term Treasury bond must have less risk than investing in a wide variety of different maturities of debt and other types of securities. Surely there is no risk involved in an investment in long-term US Treasury obligations. But consider the risks. A runaway inflation may wipe out the value of the long-term obligation; the rates of return on obligations of the Weimar Republic - surely the most secure obligations of any debt issuing entity in Germany after World War I - rose to 50 percent, 100 percent, 1000 percent, and higher and yet those who invested in them in 1919 and 1920 were completely wiped out by 1925. Even a persistent and lengthy period of "modest" inflation such as we experienced in the US in the last sixty years of the Twentieth Century ensured that most people who invested in long-term Treasury obligations got a real return that was far less than the stated or "nominal" return. Bear in mind that an inflation rate of six (6) percent compounded annually (and inflation is best considered a compound interest phenomenon) will result in the price level doubling in twelve (12) years. Many who invested in US obligations in the early '40s and in the '60s and '70s got back in purchasing power less than they invested in those obligations. (To show that government has a sense of humor, the nominal returns, the interest payments, were subject to income tax during most of this period. After tax, the return on most Treasury obligations issued during these periods was negative!) Long-term bonds are exposed to many other risks. There is, for example, reinvestment risk: when the bonds mature it may be in the midst of an interest rate climate where available interest rates for new investments are very low. It will be shown in the following paragraphs that the returns to a broadly diversified portfolio will generally be higher and the risks will be lower than the returns to a single investment in a long-term obligation no mature how secure that investment may be.
The argument will essentially follow that introduced by Harry Markowitz. Markowitz, a Nobel Laureate in Economics, is generally considered the Father of modern portfolio theory. In doing so, we shall employ some very elementary statistical measures. Please do not be intimidated! These will be introduced in the most elementary and intuitive manner and if you can compute squares and understand square roots you will have no problem with the arithmetic!
An Example
Let's consider the highly simplified example in Table 1.
| Table 1. TWO POSSIBLE INVESTMENTS |
| Stock 1 | Stock 2 |
| Possible returns | 2 or 8 | 0 or 18 |
| Probability | 0.5 each | 0.5 each |
| Expected return | 5 | 9 |
| Variance | 9 | ? |
| Standard deviation | 3 | ? |
Suppose there are only two possible investments, Stock 1 and Stock 2 and that the investor can either put all her investments in either Stock 1 or Stock 2 or can put part of her resources in one and the remainder in the other. Which is the better choice for the investor?
Let's consider first the expected return for the two investments. Stock 1 can give the investor a return of 2 percent or a return of 8 percent with an equal probability of both. The expected return from this investment is 5 percent determined as follows:
- expected return = 0.5(2) + 0.5(8) = 1 + 4 = 5
Just to ensure that you follow these numbers you should verify that the expected return for Stock 2 is 9 percent. If expected return is all that matters to our intrepid investor, clearly Stock 2 is a better bet than Stock 1.
But expected return is not everything. The return from Stock 2 could be as low as zero percent and that could be very disturbing to some investors. Suppose our investor is a retired Maine schoolteacher whose parents taught her the Golden Rule and its first corollary which is that you do not dip into capital. What will she do if the return works out to be zero percent? If she owns Stock 1 at least she can count on a return of two percent and she may be able to survive on that without having to dip into savings.
To measure the risk in this situation we might consider the spread or scatter of the possible values from the expected values. We might, for example, take the difference between the possible values and the expected values, weight them by the probability of each and sum. Unfortunately, there are some problems with this metric. For Stock 1 the calculation would be as follows:
- 0.5X(2-5) + 0.5X(8-5) = -1.5 + 1.5 = 0!
This suggests that the relevant measure of risk is zero, which clearly defies what Table 1 suggests. If you make a similar calculation for Stock 2 you will find it also works out to zero. These calculations suggest that neither stock has any risk and our Maine schoolteacher knows better! The problem is that the pluses and minuses cancel each other out. We need a measure such that the ups do not cancel out the downs.
One of the commonest ways around this problem is to square the deviations before weighting them by the probabilities and summing, a calculation which defines the variance. Here's the calculation for Stock 1:
- Variance = 0.5X(2-5)2 + 0.5X(8-5)2 = 4.5 + 4.5 = 9.
You will find it useful to figure the variance for stock 2 and when you think you have the answer run your mouse over this line.
A problem with the variance, of course, is that, being a squared dimension, it is a little difficult to compare with the unsquared numbers with which we began. This problem is solved by taking the positive square root of the variance, known as the "standard deviation." The standard deviation for Stock 1 is the square root of the variance or 3.
As is apparent from Table 1, Stock 2 has much greater risk than Stock 1. The variance and standard deviation for stock 2 are 81 and 9, respectively, as compared with a variance for Stock 1 of 9 and a standard deviation of 3. Using the variance and standard deviation, it is apparent that Stock 2 is much riskier than Stock 1.
So which is the better investment, Stock 1 or 2? Our Maine school teacher might prefer the safety and security of knowing that she will at least earn 2 percent with Stock 1; she might accordingly settle for an "expected return" of 5 percent even knowing that the expected return from Stock 2 is 9 percent. A more adventurous investor (possibly you?), one not quite so risk averse, might be willing to countenance the far greater riskiness or variability of Stock 2 in order to earn the higher expected return of 9 percent. If the only choice is to put all one's money in 1 or 2 which choice might be made will depend on the personality and risk aversion of the investor.
What will now be explored is whether it might be better to put part of the nest egg in Stock 1 and the other part in 2. Can one improve performance by diversifying?
A Simple Portfolio
The simple portfolio we shall consider involves some of the money being invested in Stock 1 and some in 2. In a sense, of course, we have already considered two simple portfolios, one the 100% in Stock 1 and the 0% in 2 that might be employed by a very risk averse individual and the 0% in 1 and 100% in 2 that might be employed by our mountain climbing friend. But the portfolio(s) now to be considered will have more than 0% and less than 100% invested in both. Some surprising results will be found for some such portfolio in terms of its ability, particularly for the acutely risk averse investor, to increase return and at the same time reduce, yes reduce, risk.
Suppose that half the school teacher's savings are invested in Stock 1 and half in Stock 2 and consider Table 2. The first column shows the various possibilities for Stock 1 and the second for 2. For example, in the first row, Stock 1 earns two percent and Stock 2 earns zero percent. In the second row, Stock 1 earns two percent and 2 earns eighteen percent and so on. In the third column, you see the return to the portfolio which is ten percent in the second row calculated as follows:
- .5X2 + .5X18 = 1 + 9 = 10.
Yes, this possibility envisages a return of two percent on Stock 1 but since only half the portfolio is in Stock 1 this contributes only one percent to the overall return on the portfolio. Likewise, Stock 2 earning eighteen percent but constituting only half the portfolio contributes only nine percent to the return on the portfolio. The total return is the one percent from Stock 1 and the nine from 2 or a total return of ten percent.
| Table 2. 50% IN STOCK 1; 50% IN STOCK 2 |
| RETURNS | PORTFOLIO |
| Stock 1 | Stock 2 | Return | Probability | Expectation |
| 2 | 0 | 1 | 0.25 | 0.25 |
| 2 | 18 | 10 | 0.25 | 2.50 |
| 8 | 0 | 4 | 0.25 | 1.00 |
| 8 | 18 | 13 | 0.25 | 3.25 |
| EXPECTED RETURN | 7.00 |
| VARIANCE | 22.5 |
| STANDARD DEVIATION | 4.74 |
You should verify that the return on the fourth row would be 13 percent. If you have any problem doing so,
run your mouse over this line.
To put this thirteen percent in the fourth row and third column of Table 2 into perspective, let's suppose our retired Maine school teacher has accumulated investment funds of $1 million and she invests $500,000 in Stock 1 and $500,000 in 2. The fourth row contemplates her earning eight percent or $40,000 (=.08X500,000) on Stock 1 and $90,000 (=.18X500,000) on Stock 2 for a total return of $130,000 (=40,000+90,000). This $130,000 in relation to her total beginning portfolion of $1 million yields the thirteen percent shown in the fourth row and third column.
To put these various possibilities together to get her expected return, we need to know something about how probable the various outcomes may be. It's been assumed that the two and the eight for Stock 1 are equally probable and likewise that the zero and eighteen are equally probable for Stock 2. Let's further assume that what happens to Stock 1 is independent of what happens to 2: if the return from Stock 1 is two percent, the return from Stock 2 is equally likely to be zero percent or eighteen percent. The combined probability of a return of two percent on Stock 1 and a return of zero percent on Stock 2 is accordingly 0.25 and there is a similar probability for each of the other three possibilities as shown in the fourth column of Table 2.
Multiplying the return in column three by the probability in column four, we get the "expectations" sometimes called "Partial expectations" in Column five. For the first row, you get a return of one multiplied by a probability of 0.25 for a partial expectation of 0.25 as shown in the final column. You might verify the 2.5 in the second row and final column and if you have any problem, run your mouse over this line.
Adding these partial expectations in the final column, you will see that the expected return from the portfolio, namely seven percent, is considerably better than if she invests only in Stock 1. Since half the portfolio is invested in Stock 2, however, presumably it is riskier than an investment only in Stock 1. The Maine school teacher can get a feel for this in the third column: the returns can be as low as one percent and as high as thirteen percent which is more than the "spread" or "range" from two to eight percent if she invests solely in Stock 1. But note that 1 to 13 as shown in the third column is nowhere near as bad as 0 to 18 if she invests solely in Stock 2.
Let's employ the variance and standard deviation (patience, dear reader, we are almost through with these complex calculations) to measure the "riskiness" of this half and half portfolio. Again, we compare the various outcomes, the 1, the 10 and so on from the third column, with the expected value for her return, the 7 in the fifth row, take the differences, square them, and weight them by their probabilities to compute the variance as follows:
- 0.25X(1-7)2+0.25X(10-7)2+0.25X(4-7)2+0.25X(13-7)2 = 9+2.25+2.25+9 = 22.5
The standard deviation is the square root of the 22.5 or 4.74.
Yes, this half and half portfolio has more risk than Stock 1 all by itself, but not much more risk. Note that she has increased her expected return to 7, she has captured half the difference between the return on Stock 1 (namely 5) and that on Stock 2 (namely 9) but the risk, as measured by the standard deviation, has only increased from 3 to 4.74. It is nowhere near the standard deviation of 9 for an investment of all her funds in Stock 2.
The Maine school teacher, no matter how risk averse she may be, may find an investment of 80 percent of her funds in Stock 1 and 20 percent in 2 clearly superior to an investment solely in Stock 1 as detailed in Table 3.
| Table 3. 80% IN STOCK 1; 20% IN STOCK 2 |
| RETURNS | PORTFOLIO |
| Stock 1 | Stock 2 | Return | Probability | Expectation |
| 2 | 0 | 1.6 | 0.25 | 0.40 |
| 2 | 18 | 5.2 | 0.25 | 1.30 |
| 8 | 0 | 6.4 | 0.25 | 1.60 |
| 8 | 18 | 10 | 0.25 | 2.50 |
| EXPECTED RETURN | 5.8 |
| VARIANCE | 9 |
| STANDARD DEVIATION | 3 |
To fully understand the attraction of this 80/20 portfolio, you ought to check a few of the calculations. For example, check the 10 return in the fourth row and third column and if you have any problem,
run your mouse over this line.Then check the 2.5 in the fourth row and final column and if you have any problem,
run your mouse over this line.
Most important of all, check the variance of 9 shown toward the bottom of the table and if you have any problem, run your mouse over this line. Note that this variance is the same, it is identical, to the 9 if she invests solely in Stock 1 and the standard deviation likewise is the same as the 3 if she invests solely in Stock 1. If she measures risk on the basis of variance or standard deviation it is apparent that this portfolio has the same risk as a portfolio consisting solely of Stock 1. Since it also has a higher return, namely 5.8 percent as compared with 5 percent for Stock 1, presumably it is a better portfolio.
Yes, we may cavil, looking at the third column, there is a possibility that her return will be only 1.6 and that is less than the 2 percent she can count on if she invests solely in Stock 1. But there is only a 0.25 probability of a return as low as 1.6 whereas there is a 0.5 possibility of such a low return with an investment solely in Stock 1. And offsetting that abysmal 1.6 return there are several other returns that are much better including a .25 probability of a return as high as 10 percent, and an investment solely in stock 1 has no possibility whatever of giving her a return of 10 percent. Maine conservatives have good values but this school teacher, who used to teach high school math, will surely prefer the 80/20 portfolio to the 100/0 portfolio.
These possibilities and other are tabulated in Table 4. You will recognize the 50/50 row as corresponding to Table 2 and the 80/20 row as corresponding to Table 3 and your calculations.
| Table 4. VARIOUS PORTFOLIOS COMPARED |
| PERCENT IN | PORTFOLIO |
| Stock 1 | Stock 2 | Return | Std Dev |
| 100 | 0 | 5 | 3 |
| 90 | 10 | 5.4 | 2.85 |
| 80 | 20 | 5.8 | 3 |
| 75 | 25 | 6 | 3.18 |
| 50 | 50 | 7 | 4.74 |
| 0 | 100 | 9 | 9 |
A particularly interesting row in Table 4 is the second or 90/10 row. Assuming you are not tired of the calculations by now, you might enjoy checking the 5.4(expected return) and 2.85 (standard deviation) in the last two columns. As compared with investing solely in Stock 1 (the first, or 100/0 row) she can, as the second row demonstrates,
boost her return from 5 to 5.4 percent and simultaneously reduce her risk as measured by the standard deviation from 3 to 2.85! Aside from the difficulty of figuring out this preferable portfolio and the tedium of cashing two sets of dividend checks, the 90/10 portfolio must be superior to the 100/0 portfolio! It has a higher return and a lower risk!
These relationships are plotted in 
Figure 1 which shows the return on the vertical axis and the risk, as measured by the standard deviation, on the horizontal axis. The labels on the curve are selected returns from Table 4. The 5 at the lower extremity of the curve, for example, represents investing 100 percent in Stock 1 and 0 percent in Stock 2, in other words, it represents not diversifying and choosing the pseudo-security of Stock 1. "Pseudo," that is, because one can have a higher return, a return of 5.8, as you demonstrated in Table 3, with no more risk as measured by the standard deviation. This superior, or "dominating", portfolio can be pictured in terms of the curve in Figure 1 as a movement directly "North" from the point marked "5" to the point marked "5.8".
And if security, as represented by the standard deviation, is the most important aspect of the investment decision, consider moving "northwest" from the point marked "5" to the point marked "5.4" which has higher return and less risk. All points on the curve from the point marked "5" to the point marked "5.4" may be said to be "inefficient" in this sense: the performance of any portfolio on this line segment may be improved in terms of return by diversifying and this improvement may be had without any increase in risk as measured by the standard deviation. In terms of the curve, take any point on the segment from 5 to 5.4 and move directly "North" from it; the point directly above it has a higher return and no more risk.
If the two stocks are the only possible investment, then the line segment from 5.4 to 9 may be referred to as representing "efficient" portfolios. This "frontier" is efficient in the sense that different positions on it may be taken by different investors with different risk-return psyches and they may do so without being obviously irrational. A very bold investor with little concern for risk might rationally invest all in Stock 2 (the point marked 9) to obtain the high expected return it promises. A very timid investor might invest all in the portfolio marked 5.4 (90/10 portfolio in the second row of Table 4) to minimize her risk. In contrast to these efficient portfolios represented by points on the line from 5.4 to 9, no portfolio represented by a point on the line from 5 to 5.4 can be considered efficient; all such portfolios may be improved in terms of returns without any increase in risk by moving "north" to points on the "efficient frontier."
An "efficient portfolio" has two characteristics. First, as explored above, the return cannot be increased without incurring additional risk. Second, and this has not been stressed above, for a given return, it is not possible to reduce the risk. For stocks with low risk and low return, it is usually true that some diversification can improve return without increasing risk.
Related Returns
An implicit assumption in the preceding analysis is that there is no relation between the returns on Stock 1 and Stock 2. A bad return for Stock 1 (the two percent return) does not change the 0.5 probability of a bad return for Stock 2 (zero percent) and does not change the 0.5 probability of a good return for Stock 2 (18 percent). (Refer back to Table 1.) Accordingly, with the probability of a bad return on Stock 1 being 0.5 and the probability of a bad return on Stock 2 also being 0.5, the probability of a bad return on both - you might call this the "joint probability" - would be 0.25, or 0.5 X 0.5. If you'll refer back to Tables 2 and 3 you'll see that the joint returns of the four possible outcomes are all 0.25. Each outcome is equally probable. Statisticians would say that the correlation between the returns on the two stocks is zero.
(Don't despair, dear reader! While it may be apparent that some statistical concepts are swirling under the surface, the approach here will be elementary and intuitive. You don't need to be a statistician to be an effective and profitable diversifier!)
Might the results have been somewhat different if there was a relationship between the returns on stocks 1 and 2? To explore, let's start with an extreme case: let's assume that when one stock is up the other is down. Suppose Stock 1 pertains to a company that produces and sells Bibles, Korans, and prayer books for agnostics. By contrast, Stock 2 represents an ownership interest in a company that produces guns, grenades and garrotes. Suppose our retired school teacher has been following the stocks of these two companies and has noticed that whenever Stock 1 has a bad year (2 percent) you can be sure that Stock 2 has a great year (18 percent) and whenever Stock 1 has a good year (8 percent) Stock 2 has a disastrous year (0 percent). She explores what would happen - Table 5 - if she put 75 percent of her savings in Stock 1 and the remaining 25 percent in Stock 2.
The first two columns, representing the return on Stocks 1 and 2 are the same as in prior Tables 2 and 3. The third column gives the return on a portfolio consisting 75 percent of Stock 1 and 25 percent of Stock 2. Thus the 1.5 in the first row may be obtained as follows:
- (0.75X2) + (0.25X0) = 1.5 + 0 = 1.5
But notice that this dreadful result cannot happen because it implies that both Stock 1 and Stock 2 have a bad year and the Maine school teacher has discovered that if one is bad the other is good. In other words, the probability of such an outcome is zero as shown in the fourth column and so the contribution to the expected value shown in the final column is zero.
| Table 5. PERFECT NEGATIVE CORRELATION |
| 75% IN STOCK 1; 25% IN STOCK 2 |
| RETURNS | PORTFOLIO |
| Stock 1 | Stock 2 | Return | Probability | Expectation |
| 2 | 0 | 1.5 | 0 | 0 |
| 2 | 18 | 6 | 0.5 | 3 |
| 8 | 0 | 6 | 0.5 | 3 |
| 8 | 18 | 10.5 | 0 | 0 |
| EXPECTED RETURN | 6 |
| VARIANCE | 0 |
| STANDARD DEVIATION | 0 |
You are encouraged to check the numbers in the second row. If you have any trouble arriving at a return of 6 in the third column, put your mouse here. Hopefully, you can easily relate to the 0.5 probability in the fourth column; we continue to assume that there is an 0.5 probability that Stock 1 will be "down" and if that is the case we know that Stock 2 is up.
Similarly, for the final column you should arrive at 3 as can readily be verified by putting your cursor here.
Note that no matter what happens, the Maine teacher will earn six percent on her money as shown in the third row from the bottom. This number is just the total of the 3 and the 3 in the final column. The variance of this portfolio is accordingly zero as you can readily verify by "mousing around" here. Likewise the standard deviation - the square root of the variance - is zero.
Admittedly, perfect negative correlation like this is difficult - impossible? - to find. If it exists, it raises the possibility of a portfolio with no variance and no standard deviation. Not all portfolios in this situation will, of course, produce zero risk. To take the two extreme cases, for example, if she invests 100 percent in Stock 1 and nothing in Stock 2, she will have a standard deviation, as we saw going all the way back to Table 1,of 3. The contrary movements of Stock 2 will not help reduce the risk of her portfolio in this case. That's the point marked "5" in 
Figure 2. Likewise if she invests 100 percent of her portfolio in Stock 2 she will have a standard deviation of 9 and extreme risk. This portfolio is represented by the point marked "9" in Figure 2. (In this case the 9 represents the return but it is also the standard deviation.) The two straight lines one going from the number "5" to the number "6" on the vertical axis and the other going from the number "6" on the vertical axis to the number "9" in the upper right corner represent the risk-return combinations that are possible in this "perfect negative correlation" scenario. The two lines representing "perfect negative correlation" (designated "R=-1" in the legend) are marked with "diamonds" in Figure 2. The point where these two line segments touch - the point marked "6" where they touch the vertical axis - is the risk return combination - the 6 percent return and the 0 standard deviation which was computed in Table 5.
Observe that the Maine teacher must like the risk return combination, the 6 percent return and the zero risk that you just computed in Table 5 better than the 5 percent return and standard deviation of 3 for an investment solely in Stock 1. She must prefer the higher return of 6 percent for the 75/25 portfolio to the lower 5 percent return for the 100/0 return for an investment solely in Stock 1. And if she is truly risk averse she must prefer the lower risk - a standard deviation of 0 - to the risk - a standard deviation of 3 - for an investment solely in Stock 1.
If, knowing that when one is up the other is sure to be down, it is possible to go further and assert that no point on the line going "northwest" from 5 to the point marked 6 on the vertical axis can be an "optimal" or "efficient" portfolio. Any portfolio represented by a point on this line segment can be improved by moving to a portfolio represented by a point directly "north" from it. Such a movement represents moving to a portfolio with a higher return and equivalent risk. For example, moving from the 100/0 portfolio represented by the point marked 5 directly north to the point marked 7 must be an improvement in the portfolio: the return is higher, 7 rather than 5 percent; and the risk, represented by a standard deviation of 3 percent is the same for both. For this situation where there is perfect negative correlation no point on the line moving northwest from 5 to 6 can be "optimal" or "efficient" to somebody who prefers a higher return and less risk. The only portfolios that can be considered "optimal" or "efficient" are those on the line segment moving northeast from the point marked 6 on the vertical axis to the point 9. These points all represent portfolios for which not more than 75 percent is invested in Stock 1. All points on the line going northwest from 5 to 6 represent portfolios with too much in Stock 1 (more than 75 percent) and not enough in Stock 2 (less than 25 percent). All such portfolios may be improved in terms of return or risk or both by moving to superior points on the line segment from 6 on the vertical axis to a point on the line from 6 on the vertical axis to 9 in the upper right corner of Figure 2.
For how many pairs of stocks is it true that the returns are perfectly negatively correlated? Probably few if any! What is the opposite extreme? Perfect positive correlation! When one is down the other is down and when one is up the other is also up.
With perfect positive correlation, it is again possible to increase return by reducing the percent of the portfolio in Stock 1 and increasing the percent in 2 but such a return improvement will always carry with it higher risk as measured by standard deviation. These tradeoffs are represented by the various portfolios on the straight line in Figure 2 heading "northeast" from the point marked "5" to the point marked "9". The line in figure 2 has little rectangles drawn on it.
You might, for example, find it interesting to verify the point at the rectangle marked 6 on this line. It represents a portfolio invested, as before, 75 percent in Stock 1 and 25 percent in 2. But now the assumption is that returns are perfectly positively correlated as in Table 6. The first three columns are identical to prior Table 5 but
| Table 6. PERFECT POSITIVE CORRELATION |
| 75% IN STOCK 1; 25% IN STOCK 2 |
| RETURNS | PORTFOLIO |
| Stock 1 | Stock 2 | Return | Probability | Expectation |
| 2 | 0 | 1.5 | 0.5 | 0.75 |
| 2 | 18 | 6 | 0 | 0 |
| 8 | 0 | 6 | 0 | 0 |
| 8 | 18 | 10.5 | 0.5 | 5.25 |
| EXPECTED RETURN | 6 |
| VARIANCE | 20.25 |
| STANDARD DEVIATION | 4.5 |
the fourth column is different. Consistent with assuming that when one is down the other is down, the probability in the first row is 0.5 and when one is up the other is up, the probability in the fourth row is likewise 0.5. The probabilities that when one is up the other is down are zero for the second and third rows. You can easily check the numbers in the last column. You will, for example, find that the variance is 20.25 and that its square root, the standard deviation, is 4.5. The return here has gone up by one quarter of the distance from 5 to 9, it has gone up to 6 percent. The risk, as measured by the standard deviation, has likewise increased by one quarter of the distance from the 3 (all in Stock 1) to 9 (all in 2), it has increased to 4.5 percent. There is a simple linear relation as shown in Figure 2.
Does diversification between the two stocks make sense for the Maine teacher? Almost undoubtedly. Only in the extreme and very rare case where there is perfect positive correlation between the returns on the two stocks - if one is down the other is down and if one is up the other is up - only in that rare (possibly non-existent) case is there a simple linear trade-off between return and risk. At the other extreme, if there is perfect negative correlation between the returns then by diversifying it is possible to boost return above the return on the stock with the lowest return and at the same time to reduce risk.
Most pairs of stocks on the market will be between these two extremes. There may be an element of return that is common to most stocks. To the extent the overall market goes up then many pairs of stocks will also go up. There is a market or "systematic" risk that doubtless relates to many pairs of stocks. But there is also an element in the return for an individual stock that is peculiar to that stock and that is uncorrelated to the market in general. There are some events that will cause the price of GM stock to falter but which will have no implications for MicroSoft stock. Many such pairs of stocks will have an element in their return which is uncorrelated to the market in general and uncorrelated to the return on other stocks in the market. To this extent, many pairs of stocks will have returns - or at least a portion of their returns - which has zero correlation as was investigated in Table 4 and Figure 1. To the extent the returns are uncorrelated, the unfavorable developments pertaining to one may be offset by favorable developments for the other. These returns form an intermediate case between the two extremes depicted in Figure 2. These uncorrelated returns account for the possibility of getting favorable effects from diversification as illustrated in Figure 1 and surrounding discussion.
Several Stocks
The market, of course, is not composed of two stocks but of a multitude. Instead of combining two stocks in her portfolio, the Maine school teacher can combine three, four, perhaps 30 or 40. With many different securities and investments included in her portfolio, much of the risk and variability relating to one stock in the portfolio will be eliminated or "drowned out" by the contrary movements of other stocks in the portfolio.
Let us start by again examining a portfolio of only two securities, Stock 2 and Stock 3. Stock 2 will be the same one used in previous illustrations but let's add another stock, Stock 3, which has the same return and risk profile as Stock 2; that is, both have possible returns of 0 or 18, the probability of each such return is 0.5, both have an expected return of 9 and both have a standard deviation of 9. Can there be any advantage from the standpoint of reducing risk of combining two such stocks in a portfolio? Did the risk reduction possibilities for the two stocks previously considered depend on combining a stock with high risk (Stock 2) with a stock with low risk (1)? It can easily be seen that risk reduction possibilities also arise for two stocks with similar risk (Stocks 2 and 3) provided their returns are not highly correlated.
If the returns of the two are perfectly correlated, then indeed there will be no advantage from the standpoint of risk reduction, of combining these two in a portfolio. The expected return of the portfolio will be 9 and the standard deviation will be 9 just as if the portfolio consisted of only one of these stocks. As suggested already, however, perfect correlation is something that exists only in the textbooks and not in the real world.
Suppose there is no correlation between the returns on the two stocks. The various possible returns for Stocks 2 and 3 are arrayed in the first two columns of Table 7. As in prior Table 2, it is assumed that the portfolio is invested equally in both, although now it is Stocks 2 and 3 rather than 1 and 2.
| Table 7. 50% IN STOCK 2; 50% IN STOCK 3 |
| RETURNS | PORTFOLIO |
| Stock 2 | Stock 3 | Return | Probability | Expectation |
| 0 | 0 | 0 | 0.25 | 0 |
| 0 | 18 | 9 | 0.25 | 2.25 |
| 18 | 0 | 9 | 0.25 | 2.25 |
| 18 | 18 | 18 | 0.25 | 4.5 |
| EXPECTED RETURN | 9 |
| VARIANCE | 40.5 |
| STANDARD DEVIATION | 6.36 |
If the returns on the two are unrelated, if there is no correlation between them, then each of the four possibilites has an equal probability of 0.25 as shown in the fourth column. Consider the fourth row: Stock 2 has a return of 18; Stock 3 has a return of 18; the expected return for the portfolio is accordingly 18 (namely 0.5X18 + 0.5X18); and multiplying this portfolio return by the probability of 0.25 we arrive at the "partial expectation" of 4.5 in the final column. Adding the various partial expectations together gives the expected return of 9 for the portfolio. The variance of
40.5, however, is only half as great as the variance of 81 if the entire portfolio consists of only one of the two stocks. The standard deviation has been reduced from 9 to 6.36, a reduction of almost one third. Risk reduction possibilities are very much available even from combining two high risk stocks like 2 and 3 provided their returns do not have a high positive correlation!
Are further risk reductions possible from adding a third stock? You can prepare a tabulation and a computation along the lines of Table 7 that will indeed demonstrate the risk reduction possibilities. Your table will have entries in eight rows instead of four. For example, the first column for Stock 2 will have the entries 0,0,0,0,18,18,18,18, the second column for Stock 3 will have the entries 0,0,18,18,0,0,18,18 and the third column for Stock 4 will have 0,18,0,...18. The fourth column containing the portfolio returns will be 0,6,6,...18. The probabilities, assuming each event is equally probable and that there is no correlation between the events, will be 0.125 in each row. The partial expectations in the final column will be 0,0.75,0.75,...2.25. Adding these partial expectations together it will be seen that the expected return for the portfolio is 9. This should be no surprise as it is the return resulting from combining three securities each of which had an expected return of 9. The variance, computed as follows:
- .125X(0-9)2 + .125X(6-9)2+...+.125X(18-9)2
- = 10.125 + 1.125 + ... + 10.125
- = 27
is only 27! (There is, dear reader, a shortcut to this 27 involving none of the tedium of enumerating the possibilities and carrying out these computations. Hopefully, you will take our word for it rather than carrying out all the computations.) Let's summarize. If the portfolio consists of putting all one's resources in one of these three stocks the variance is 81. Assuming the return on each stock is independent of the others (i.e. correlation of 0) and by investing one's funds equally in each of the three securities the variance may be reduced to one third of 81, namely 27.
Would you care to speculate if a fourth stock - a stock with exactly the same profile as stock 2 is added to the portfolio? Yes, the variance of the portfolio will be reduced to one quarter, namely 20.25.
A fifth stock reduces the variance to a fifth, namely 16.2.
How about a portfolio of 100 stocks, all like Stock 2 with a variance of 81 and all uncorrelated with each other? Yes, the variance will be reduced to one hundredth, or 0.81.
| Table 8. MULTIPLE STOCKS LIKE STOCK 2 |
| PORTFOLIO EVENLY DIVIDED |
| Number of Stocks | Variance | Standard Deviation |
| 1 | 81 | 9 |
| 2 | 40.5 | 6.36 |
| 3 | 27 | 5.20 |
| 4 | 20.25 | 4.50 |
| ... | ... | ... |
| 100 | 0.81 | 0.9 |
The Maine teacher has found Heaven wrapped inside Nirvana and embroidered with Utopia! All she has to do is find several stocks with the characteristics of Stock 2 and divide her savings equally among the different stocks and she can be assured of a high expected return, a return of 9, with, as summarized in Table 8, practically no risk. If she is willing to put in the time and effort to fill in deposit tickets
for the dividends resulting from 100 different stocks, she can reduce the standard deviation to 0.9. She doesn't need Stock 1 with its low expected return to reduce her standard deviation to tolerable levels. All she needs is enough stocks like Stock 2 to almost completely eliminate risk!